Question

How do you use normal distribution to approximate the binomial distribution ?

Answer 1

Before going further, we should know what normal distribution and binomial distribution mean. A normal distribution, sometimes called the bell curve, is a distribution that occurs naturally in many situations. For example, the bell curve is seen in tests like SAT where the bulk of students will score the average grade C while a smaller number of students will score a B or D . An even smaller percentage of students score an A or F. The curve is symmetrical.

Complete step by step solution:
We know that normal distribution is also called a bell curve. And it is mostly observed in real life where we have seen the example of a common admission test that many students take. The grade of all the students makes up and forms the bell curve.
The mean ,mode, median of normal distribution are equal and the area of the bell curve is always $1$.
Binomial distribution can be thought of as simply the probability of success or failure outcome in an experiment or survey that is repeated multiple times. The binomial is a type of distribution with two possible outcomes which is exactly why we have the prefix “bi”.
The two possible outcomes which come from our binomial distribution is either success or failure. For example, when a coin is tossed multiple times, the two outcomes which come out of tossing It either heads or tails. Likewise, sitting for a test also has only two outcomes which are either pass or fail.
We use normal distribution to approximate binomial distribution in the following :
The normal distribution can be used as an approximation to the binomial distribution, under certain circumstances, namely :
If $X\sim B\left( n,p \right)$ and if $n$ is large and/or $p$ is close to $\dfrac{1}{2}$, then $X$ is approximately$N\left( np,npq \right)$
Where $n$ is the number of times we have performed an event,$p$ is the probability of the success, $q$ is the probability of failure and $q=1-p$.
In some cases, working out a problem using the Normal distribution may be easier than using a Binomial.

Note: Statistics is a very important chapter. There are a lot of formulas that are needed to memorize so as to be able to solve a question quickly during the exam. Lengthy calculations are also involved in problems of Statistics. We should practice a lot of problems and try to reduce the calculating errors that we are getting so as to get at least a precise answer in the exam. Connecting each topic of Statistics to real life examples will help make the concept clear and understand it’s application.